In all the previous developments we had followed the “bottom-up” approach, i.e. we uncovered the details of the finite noncommutative geometry \(({\mathcal {A}}_F,{\mathcal {H}}_F,D_F)\)  from the experimental information contained in the standard model coupled to gravitation. In 2014,
in collaboration with S. Mukhanov [1]}, [2]} we were investigating the purely geometric problem of encoding 4-manifolds in the most economical manner in the spectral formalism. The problem had no a priori link with the standard model of particle and forces and the idea was to treat the coordinates in the same way as the momenta are assembled together in a single operator using the gamma matrices. The great surprise was that this investigation gave the conceptual explanation of the finite noncommutative geometry from Clifford algebras! This is described in details in [3]} to which we refer. What we obtained is a higher form of the Heisenberg commutation relations between \(P\)  and \(Q\) , whose irreducible Hilbert space representations correspond to 4-dimensional spin geometries. The role of \(P\)  is played by the
Dirac operator as explained before, and the role of \(Q\)  by the Feynman slash of coordinates using Clifford algebras. The proof that all spin geometries are obtained relies on deep results of immersion theory and ramified coverings of the sphere. The volume of the 4-dimensional geometry is automatically quantized by the index theorem and the spectral model, taking into account the inner automorphisms due to the noncommutative nature of the Clifford algebras, gives Einstein gravity coupled with the slight extension of the Standard Model, as a Pati–Salam model. This model was shown in [4]}, [5]} to yield unification of coupling constants. We refer to the survey [6]} for a concise account of the whole story of the evolution of this theory from the early days to now.
<TABLE>